Philosophy of Geometry from Riemann to Poincaré

472 Pages·1978·23.889 MB·English

by  Roberto Torretti (auth.)

PHILOSOPHY OF GEOMETRY FROM RIEMANN TO POINCARE
A PALLAS PAPERBACK
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ROBERTO TORRETTI
University ofP uerto Rico
PHILOSOPHY OF
GEOMETRY
FROM RIEMANN
TO POINCARE
D. REIDEL PUBLISHING COMPANY
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A MEMBER OF THE KLUWER " ACADEMIC PUBLISHERS GROUP
DORDRECHT/BOSTON/LANCASTER
Library of Congress Cataloging in Publication Data
Torretti, Roberto, 1930-
Philosophy of geometry from Riemann to Poincare.
(Episteme; v. 7)
Includes bibliographical-Ieferences and index.
1. Geometry-Philosophy. 2. Geometry-History.
I. Title.
QA447.T67 516'.001 78-12551
ISBN-13: 978-90-277-1837-2 e-ISBN-13: 978-94-009-9909-1
DOl: 10.1007/978-94-009-9909-1
Published by D. Reidel Publishing Company,
P.O. Box 17,3300 AA Dordrecht, Holland.
Sold and distributed in the U.S.A. and Canada
by K1uwer Academic Publishers,
190 Old Derby Street, Hingham, MA 02043, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, Holland.
First published in J 978 in hardbound edition by Reidel
in the series Episteme
All Rights Reserved
© 1978, 1984 by D. Reidel Publishing Company, Dordrecht, Holland
and copyrightholders as specified on the appropriate pages within
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
To Carla
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T ABLE OF CONTENTS
PREFACE xi
CHAPTERljBACKGROUND 1
1.0.1 Greek Geometry and Philosophy 1
1.0.2 Geometry in Greek Natural Science 10
1.0.3 Modern Science and the Metaphysical Idea of Space 23
1.0.4 Descartes' Method of Coordinates 33
CHAPTER 2jNON-EUCLIDEAN GEOMETRIES 40
2.1 PARALLELS 41
2.1.1 Euclid's Fifth Postulate 41
2.1.2 Greek Commentators 42
2.1.3 Wallis and Saccheri 44
2.1.4 Johann Heinrich Lambert 48
2.1.5 The Discovery of Non-Euclidean Geometry 53
2.1.6 Some Results of Bolyai-Lobachevsky
Geometry 55
2.1.7 The Philosophical Outlook of the Founders of
Non-Euclidean Geometry 61
2.2 MANIFOLDS 67
2.2.1 Introduction 67
2.2.2 Curves and their Curvature 68
2.2.3 Gaussian Curvature of Surfaces 71
2.2.4 Gauss' Theorema Egregium and the Intrinsic
Geometry of Surfaces 76
2.2.5 Riemann's Problem of Space and Geometry 82
2.2.6 The Concept of a Manifold 85
2.2.7 The Tangent Space 88
2.2.8 Riemannian Manifolds, Metrics and Curvature 90
2.2.9 Riemann's Speculations about Physical Space 103
2.2.10 Riemann and Herbart. Grassmann 107
vii
viii T ABLE OF CONTENTS
2.3 PROJECTIVE GEOMETRY AND PROJECTIVE METRICS 110
2.3.1 Introduction 110
2.3.2 Projective Geometry: An Intuitive Approach 111
2.3.3 Projective Geometry: A Numerical Interpretation 115
2.3.4 Projective Transformations 120
2.3.5 Cross-ratio 124
2.3.6 Projective Metrics 125
2.3.7 Models 132
2.3.8 Transformation Groups and Klein's Erlangen
Programme 137
2.3.9 Projective Coordinates for Intuitive Space 143
2.3.10 Klein's View of Intuition and the Problem of
Space-Forms 147
CHAPTER 3/ FOUNDATIONS 153
3.1 HELMHOLTZ'S PROBLEM OF SPACE 155
3.1.1 Helmholtz and Riemann 155
3.1.2 The Facts which Lie at the Foundation of Geometry 158
3.1.3 Helmholtz's Philosophy of Geometry 162
3.1.4 Lie Groups 171
3.1.5 Lie's Solution of Helmholtz's Problem 176
3.1.6 Poincare and Killing on the Foundations of
Geometry 179
3.1.7 Hilbert's Group-Theoretical Characterization of
the Euclidean Plane 185
3.2 AXIOMATICS 188
3.2.1 The Beginnings of Modern Geometrical
Axiomatics 188
3.2.2 Why are Axiomatic Theories Naturally Abstract? 191
3.2.3 Stewart, Grassmann, Plucker 199
3.2.4 Geometrical Axiomatics before Pasch 202
3.2.5 Moritz Pasch 210
3.2.6 Giuseppe Peano 218
3.2.7 The Italian School. Pieri. Padoa 223
3.2.8 Hilbert's Grundlagen 227
3.2.9 Geometrical Axiomatics after Hilbert 239
3.2.10 Axioms and Definitions. Frege's Criticism of
Hilbert 249
TABLE OF CONTENTS ix
CHAPTER 4/ EMPIRICISM, APRIORISM,
CONVENTIONALISM 254
4.1 EMPIRICISM IN GEOMETRY 256
4.1.1 John Stuart Mill 256
4.1.2 Friedrich Ueberweg 260
4.1.3 Benno Erdmann 264
4.1.4 Auguste CaHnon 272
4.1.5 Ernst Mach 278
4.2 THE UPROAR OF BOEOTIANS 285
4.2.1 Hermann Lotze 285
4.2.2 Wilhelm Wundt 291
4.2.3 Charles Renouvier 294
4.2.4 Joseph Delboeuf 298
4.3 RUSSELL'S APRIORISM OF 1897 301
4.3.1 The Transcendental Approach 301
4.3.2 The 'Axioms of Projective Geometry' 303
4.3.3 Metrics and Quantity 307
4.3.4 The Axiom of Distance 309
4.3.5 The Axiom of Free Mobility 314
4.3.6 A Geometrical Experiment 318
4.3.7 Multidimensional Series 319
4.4 HENRI POINCARE 320
4.4.1 Poincare's Conventionalism 320
4.4.2 Max Black's Interpretation of Poincare's
Philosophy of Geometry 325
4.4.3 Poincare's Criticism of Apriorism and Empiricism 328
4.4.4 The Conventionality of Metrics 335
4.4.5 The Genesis of Geometry 340
4.5.6 The Definition of Dimension Number 352
APPENDIX 359
1. Mappings 359
2. Algebraic Structures. Groups 359
3. Topologies 360
4. Differentiable Manifolds 361
x TABLE OF CONTENTS
NOTES 375
To Chapter 1 375
To Chapter 2 379
Part 2.1 379
Part 2.2 382
Part 2.3 386
To Chapter 3 391
Part 3.1 391
Part 3.2 395
To Chapter 4 403
Part 4.1 403
Part 4.2 407
Part 4.3 409
Part 4.4 412
REFERENCES 420
INDEX 440
PREFACE
Geometry has fascinated philosophers since the days of Thales and
Pythagoras. In the 17th and 18th centuries it provided a paradigm of
knowledge after which some thinkers tried to pattern their own
metaphysical systems. But after the discovery of non-Euclidean
geometries in the 19th century, the nature and scope of geometry
became a bone of contention. Philosophical concern with geometry
increased in the 1920's after Einstein used Riemannian geometry in
his theory of gravitation. During the last fifteen or twenty years,
renewed interest in the latter theory - prompted by advances in
cosmology - has brought geometry once again to the forefront of
philosophical discussion.
The issues at stake in the current epistemological debate about
geometry can only be understood in the light of history, and, in fact,
most recent works on the subject include historical material. In this
book, I try to give a selective critical survey of modern philosophy of
geometry during its seminal period, which can be said to have begun
shortly after 1850 with Riemann's generalized conception of space
and to achieve some sort of completion at the turn of the century with
Hilbert's axiomatics and Poincare's conventionalism. The philosophy
of geometry of Einstein and his contemporaries will be the subject of
another book.
The book is divided into four chapters. Chapter 1 provides back
ground information about the history of science and philosophy.
Chapter 2 describes the development of non-Euclidean geometries
until the publication of Felix Klein's papers 'On the So-called Non
Euclidean Geometry' in 1871-73. Chapter 3 deals with 19th-century
research into the foundations of geometry. Chapter 4 discusses
philosophical views about the nature of geometrical knowledge from
John Stuart Mill to Henri Poincare.
Modern philosophy of geometry cannot be separated from in
vestigations concerning fundamental geometrical concepts which
have been conducted by professional mathematicians in what are
xi